In numerous applications it is necessary to process time-varying three-component signals for analysis and filtering based on polarization properties. Some examples are measurements of time-varying electric and magnetic fields, and three-component seismograms.
The Fourier transform of a function has served as the most important transform in numerous signal processing applications and is still widely used in signal analysis. Standard Fourier analysis reveals individual frequency components involved in a signal. However, in many situations where frequencies change with time or space the standard Fourier analysis does not provide sufficient information. In numerous applications, a change of frequencies with time or space reveals important information.
Time-Frequency Representations (TFRs) are capable of localizing spectra of events in time, thus, overcoming the deficiency of the standard Fourier analysis and providing a useful tool for signal analysis in numerous applications. One such TFR is the Stockwell-transform (S-transform) disclosed in: Stockwell R. G., Mansinha L., Lowe R. P., “Localization of the complex spectrum: the S-transform”, IEEE Trans. Signal Process, 1996; 44, 998-1001. The S-transform is a spectral localization transform that utilizes a frequency adapted Gaussian window to achieve optimum resolution at each frequency. Analogously, the S-transform is also capable of localizing spectra of events in space.
An nth complex coefficient of a Discrete Fourier Transform (DFT) of each individual component of a three-component signal gives amplitude and phase of a one-dimensional sinusoidal oscillation whose frequency is determined by n. When the DFT's of all three components are considered together, their nth complex coefficients provide amplitudes and phases of three sinusoidal oscillations in x, y, and z directions of a 3D space having the same frequency, resulting in an elliptically polarized motion in the 3D space. Analysis of the elliptically polarized motion reveals important polarization dependent information of the three-component signal.
It would be advantageous to provide a signal processing method and system for polarization analysis and polarization filtering of time-varying three-component signals based on the elliptically polarized motion in the 3D space and, in particular, based on polarization ellipses that depend on both time and frequency.